Fast reaction limit and forward-backward diffusion: a Radon-Nikodym approach

TL;DR

This paper refines the analysis of singular limits in reaction-diffusion equations using Radon-Nikodym theory, providing sharper conditions and characterizations for oscillation behavior in forward-backward diffusion models.

Contribution

It introduces a Radon-Nikodym approach to improve the understanding of oscillations in singular limits, simplifying conditions and extending results to broader cases.

Findings

Sharper characterization of Young measures in reaction limits

Simplified conditions for oscillation behavior in piecewise affine cases

Validation of results through numerical simulations

Abstract

We consider two singular limits: fast reaction limit with nonmonotone nonlinearity and regularization of forward-backward diffusion equation. It was proved by Plotnikov that for cubic-type (nondegenerate) nonlinearities, the limit oscillates between at most three states. In this paper we make his argument more optimal and we sharpen the previous result: we use Radon-Nikodym theorem to obtain a pointwise identity characterizing the Young measure. As a consequence, we establish a simpler condition which implies Plotnikov result for piecewise affine functions. We also prove that the result is true if the Young measure is not supported in the so-called unstable zone, the fact observed in numerical simulations.